Lipschitz regularity of minimizers of variational integrals with variable exponents

Abstract

In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form F(v,Ω)=∫ΩF(x,Dv(x))dx,where, for n≥2 and N≥1, Ω is a bounded open set in Rn, u∈W1,1(Ω,RN) and the energy density F:Ω×RN×n→R satisfies the so called variable growth conditions. The main novelty here is that we assume an almost critical regularity in the Orlicz Sobolev setting for the energy density as a function of the x variable.